Title: The Story of Wavelets Theory and Engineering Applications
1The Story of WaveletsTheory and Engineering
Applications
Jamboree 4 February 7, 2001
- In todays show
- Time frequency representation
- Instantaneous frequency and group delay
- Short time Fourier transform Analysis
- Short time Fourier transform Synthesis
- Discrete time STFT
2Time Frequency Representation
- Why do we need it?
- Time info difficult to interpret in frequency
domain - Frequency info difficult to interpret in time
domain - Perfect time info in time domain , perfect freq.
info in freq. domain Why? - How to handle non-stationary signals
- Instantaneous frequency
- Group Delay
3Instantaneous Frequency Group Delay
- Instantaneous frequency defined as the rate of
change in phase - A dual quantity group delay defined as the rate
of change in phase spectrum
Frequency as a function of time
Time as a function of frequency
What is wrong with these quantities???
4Time Frequency Representation in Two-dimensional
Space
TFR
Linear STFT, WT, etc.
Non-Linear
Quadratic Spectrogram, WD
5The Short Time Fourier Transform
- Take FT of segmented consecutive pieces of a
signal. - Each FT then provides the spectral content of
that time segment only - Spectral content for different time intervals
- ?Time-frequency representation
Time parameter
Signal to be analyzed
FT Kernel (basis function)
Frequency parameter
STFT of signal x(t) Computed for each window
centered at t? (localized spectrum)
Windowing function (Analysis window)
Windowing function centered at t?
6Properties of STFT
- Linear
- Complex valued
- Time invariant
- Time shift
- Frequency shift
- Many other properties of the FT also apply.
7Alternate Representation of STFT
STFT The inverse FT of the windowed spectrum,
with a phase factor
8Filter Interpretation of STFT
X(t) is passed through a bandpass filter with a
center frequency of Note that ?(f) itself is a
lowpass filter.
9Filter Interpretation of STFT
X
x(t)
10Time-Frequency Resolution
- Closely related to the choice of analysis window
- Narrow window ? good time resolution
- Wide window (narrow band) ? good frequency
resolution - Two extreme cases
- ?(T)?(t)? excellent time resolution, no
frequency resolution - ?(T)1? excellent freq. resolution (FT), no time
info!!! - How to choose the window length?
- Window length defines the time and frequency
resolutions - Heisenbergs inequality
- Cannot have arbitrarily good time and frequency
resolutions. One must trade one for the other.
Their product is bounded from below.
11Time-Frequency Resolution
Frequency
Time
12Time Frequency Signal Expansion and STFT
Synthesis
Basis functions
Coefficients (weights)
Synthesis window
Synthesized signal
- Each (2D) point on the STFT plane shows how
strongly a time - frequency point (t,f) contributes to the signal.
- Typically, analysis and synthesis windows are
chosen to be identical.
13STFT Example
300 Hz 200 Hz 100Hz 50Hz
14STFT Example
15STFT Example
a0.01
16STFT Example
a0.001
17STFT Example
a0.0001
18STFT Example
a0.00001
19Discrete Time Stft
20Homework
Write a program that computes the STFT of a real
signal. Use the Gaussian window as an analysis
window. Make a, T, F variable. Extra credit
Implement the Inverse STFT